Brazilian Journal of Physics, vol. 34, no. 3A, September, 2004 875
Topological Charge Screening and Pseudoscalar Glueballs
Hilmar Forkel
IFT - Universidade Estadual Paulista, Rua Pamplona, 145, 01405-900 S ˜ao Paulo, SP, Brazil
and
Institut f¨ur Theoretische Physik, Universit¨at Heidelberg, D-69120 Heidelberg, Germany
Received on 6 October, 2003
Topological charge screening in the QCD vacuum is found to provide crucial nonperturbative contributions to the short-distance expansion of the pseudoscalar (0−+) glueball correlator. The screening contributions enter
the Wilson coefficients and are an indispensable complement to the direct instanton contributions. They re-store consistency with the anomalous axial Ward identity and remedy several flaws in the0−+glueball sum
rules caused by direct instantons in the absence of screening (lack of resonance signals, violation of the pos-itivity bound and of the underlying low-energy theorem). The impact of the finite width of the instanton size distribution and the (gauge-invariant) renormalization of the instanton contributions are also discussed. New predictions for the0−+glueball mass and decay constant are presented.
1
Introduction
The glueball sector of QCD has remained intriguing and challenging since the early days of QCD [1] and glueball physics offers promising opportunities for the study of low-energy gluon dynamics and of the often elusive gluonic component in hadron wavefunctionals. One such oppor-tunity was recently exploited in OCD sum rule analyses, which found nonperturbative short-distance physics in the form of direct instantons [2] to play a crucial role in the structure and dynamics of the scalar (0++) glueball [3, 4].
Indeed, the instanton-improved operator product expansion (IOPE) of the 0++ glueball correlator resolves two
long-standing problems of the conventional sum rules (the mu-tual inconsistency of different Borel moment sum rules and the conflict with the underlying low-energy theorem [5, 6]), generates new scaling relations between fundamental glue-ball and instanton properties, and leads to improved sum rule predictions for scalar glueball properties [3]. (See also the subsequent gaussian sum rule analysis [7], based on the same instanton contributions.) The Borel sum rule analy-sis has recently been improved and extended (to realistic in-stanton size distributions and renormalized inin-stanton contri-butions) in [4]. Hard nonperturbative contributions to the pseudoscalar glueball IOPE and sum rules were also inves-tigated. On these we will report in the following. A com-prehensive analysis of both spin-0 glueball channels can be found in Ref. [4].
The implementation of the direct instanton contributions in the0−+channel is straightforward: the expressions from
the 0++ channel [3] can simply be taken over, with their
signs inverted. This is a consequence of the (Minkowski) (anti-) self-duality of the (anti-) instanton’s field strength,
G(µνI,I¯) = ±iG˜( I,I¯)
µν . However, one immediately suspects
that adding the dominant and, due to the sign change, strongly repulsive instanton contributions will seriously un-balance the Borel sum rules. Two of the consequences were recently observed in [8]: any reliable signal for a pseu-doscalar glueball resonance disappears (in contradiction to lattice evidence), and even the fundamental spectral positiv-ity bound is violated. In addition, the crucial low-energy the-orem for the zero-momentum0−+ glueball correlator, and
therefore the underlying anomalous axial Ward identity, is strongly violated [4]. As pointed out in [4], these problems have an appealing solution in the form of additional nonper-turbative short-range contributions to the IOPE, associated with topological charge screening. Below, we will sketch the implementation of the screening contributions and their impact on the sum rule analysis. We also comment on the effects of realistic instanton size distributions and the renor-malization of the instanton-induced Wilson coefficients.
2
Correlator, IOPE and sum rules
Our discussion will be based on the pseudoscalar glueball correlation function
ΠP(x) =h0|T OP(x)OP(0)|0i, (1)
whereOP is the standard gluonic interpolating field
OP(x) =αsGµνa (x) ˜Gaµν(x) (2)
(G˜µν ≡(i/2)εµνρσGρσ), and its Fourier transform
ΠP(−q2) =i
Z
876 Hilmar Forkel
The zero-momentum limit of this correlator is governed by the low-energy theorem (LET) [9]
ΠP¡q2= 0¢= (8π)2
mumd
mu+mdh
¯
qqi (4)
(for three light flavors and mu,d ≪ ms) which derives
from the axial anomaly and imposes stringent consistency requirements on the sum rule analysis [4].
The hadronic information in the glueball correlators is most directly accessible in the dispersive representation
ΠP¡Q2¢=
1
π Z ∞
si
dsIm ΠP(−s)
s+Q2 (5)
where the necessary number of subtractions is implied. The QCD sum-rule description of the spectral functions contains one or two resonances (in zero-width approximation) and the local-duality continuum, i.e.
Im Π(Pph)(s) =π
2
X
i=1
fP i2 m4P iδ
¡
s−m2P i
¢
+θ(s−s0) Im Π(PIOP E)(s) (6)
The continuum representation covers the invariant-mass re-gion ”dual” to higher-lying resonances and multi-hadron continuum, starting at an effective threshold s0. It is
ob-tained from the discontinuities of the IOPE, i.e. the expan-sion of the correlator at large, spacelike momenta Q2 ≡
−q2≫ΛQCDinto condensates of operatorsOˆDof
increas-ing dimensionD, ΠP(Q2) =
X
D=0,4,...
˜
CD(P)¡
Q2;µ¢D
0¯¯ ¯OˆD
¯ ¯ ¯0
E
µ . (7)
“Hard” field modes with momenta|k|> µcontribute to the momentum-dependent Wilson coefficientsC˜D¡Q2¢while
“soft” modes with|k| ≤µgenerate the condensatesDOˆD
E
. In order to write down the sum rules, the IOPE - with the continuum subtracted and weighted by powers of−Q2
-is Borel-transformed,
R(P,kIOP E)(τ;s0) = ˆB
" ¡
−Q2¢k
π
Z s0
0
dsIm Π
(IOP E)
P (−s)
s+Q2
#
(8) =−δk,−1Π(PIOP E)(0)
+1
π Z s0
0
dsskIm Π(IOP E)
P (s)e
−sτ,
fork≥ −1. The hadronic parametersmP i, fP i, s0are then
determined by matching these moments in the fiducial τ -region to their resonance-induced counterparts (and a sub-traction constant fork = −1, fixed by the LET (4)). The resulting IOPE sum rules are
R(P,kIOP E)(τ;s0) = 2
X
i=1
fP i2 m4+2P i ke
−m2 P iτ−δ
k,−1Π(Pph)(0).
(9)
The perturbative contributions to the IOPE coefficients, i.e. the conventional OPE, can be found in [10, 11, 8, 4]. In the following we will focus on the nonperturbative contribu-tions due to direct instantons and topological charge screen-ing.
3
Direct instantons
Dominant direct instanton contributions to the0−+glueball
correlator are received by the unit-operator IOPE coefficient, ˜
C(P,I+ ¯I)D
0 = Π
(I+ ¯I)
P . They are best calculated inx-space,
with the result ˜
C(P,I+ ¯I)D
0
¡ x2¢
=−2
83
7
Z
dρn(ρ) 1
ρ4 2F1
µ
4,6,9
2,−
x2
4ρ2
¶ .
(10) The further evaluation requires the (anti-) instanton distri-butionnI,I¯(ρ)with its two leading moments, the instanton
density¯nand average sizeρ¯, as input. All previous studies of direct instanton effects have relied on the simplest possi-ble, spike-like approximationn(ρ) = ¯nδ(ρ−ρ¯). In Ref.[4] a realistic finite-width distribution was implemented instead, which is fully determined by¯n,ρ¯and the known small- and large-ρbehavior [12], (forNc =Nf = 3)
ng(ρ) = 2
18
36π3
¯ n ¯ ρ µ ρ ¯ ρ ¶4 exp µ − 2 6
32π
ρ2
¯
ρ2
¶
. (11)
From the Fourier transform of (10) one finds the direct-instanton-induced Borel moments [3]
L(kI−+ ¯1I)(τ) =−2
6π2−∂k
∂τk
Z
dρn(ρ)x2e−x
× ·
(1 +x)K0(x) +
µ
2 +x+2
x ¶
K1(x)
¸ ,
(12) (x=ρ2/2τ, k≥0) and from the imaginary part [3]
Im Π(PI+ ¯I)(−s) =−24π4 Z
dρn(ρ)ρ4s2J2¡√sρ¢Y2¡√sρ¢
(13) at timelike momenta one then has
R(P,kI+ ¯I)(τ) =−27π2δk,−1
Z
dρn(ρ)−24π3 Z
dρ
×n(ρ)ρ4 Z s0
0
dssk+2J2¡√sρ¢Y2¡√sρ¢e−sτ.
(14) The direct-instanton contributions (with subtracted contin-uum) are an important complement to the corresponding perturbative ones.
The realistic (finite-width) instanton size distribution tames the rising oscillations of the imaginary part (13) at large s(an artefact with misleading impact on the s0
de-pendence of the moments) into a strong decay ∝ s−5/2.
Brazilian Journal of Physics, vol. 34, no. 3A, September, 2004 877
of the instanton-induced coefficients by excluding contribu-tions from instantons with sizeρ > µ−1, i.e. by replacing
n(ρ)→n˜µ(ρ)≡θβ
¡
ρ−µ−1¢
n(ρ) (15)
with a “soft” step function θβ (e.g. θβ¡ρ−µ−1¢ =
£
exp¡ β¡
ρ−µ−1¢¢
+ 1¤−1
). The instanton-induced µ -dependence turns out to be relatively weak forµ <ρ¯−1, in
complicance with the other sources ofµdependence. Note that the standard spike distribution (withρ < µ¯ −1) misses
the reduction of the total instanton density to the direct in-stanton part,
¯
n=
Z ∞
0
dρn(ρ)→ Z ∞
0
dρn˜µ(ρ)≡n¯direct. (16)
4
Topological charge screening
As argued above, it is not surprising that the dominant and strongly repulsive direct instanton contributions, when added as the sole nonperturbative contributions to the IOPE coefficients, upset the0−+glueball sum rules and have the
mentioned, detrimental effects. In fact, this suggests that additional important contributions, which should predomi-nantly affect the pseudoscalar IOPE, are still amiss. And in-deed, the0−+glueball correlator is proportional to the
topo-logical charge correlators and therefore maximally sensitive to the short-distance topological charge (probably mainly instanton - antiinstanton) correlations in the QCD vacuum. Their impact on the pseudoscalar glueball correlators was found to be exceptionally strong in the instanton liquid model [2, 13]. Topological charge correlations are created by light-quark loops or, equivalently at low energies, by the attractive (repulsive)η′
-meson exchange forces between opposite-sign (equal-sign) topological charges. They lead to Debye screening of the topological charge with “screening mass”mη′and the corresponding (small!) screening length
λD ∼m
−1
η′ ∼0.2fm [14, 15]. Sincemη′ > µ, the screen-ing correlations contribute to the Wilson coefficients of the pseudoscalar glueball correlator.
The screening contributions can be obtained from the coupling of theη′
mesons to the topological charge density, as dictated by the axial anomaly [16]. In the chiral limit, whereη′
=η0is purely flavor-singlet, and fork.mη′the coupling to the topological charges in the vacuum medium (approximated for simplicity as concentrated in pointlike in-stantons) is governed by the effective lagrangian [14, 15]
L= 1
2(∂η0)
2
−2¯ncos (γη0η0+θ) (17)
wheren¯is the global topological charge density and where we have introduced a source θ(x)for the local topologi-cal charge densityQ(x). Note the screening massm2
scr =
2¯nγ2
η0 of theη0. Taking two derivatives of the
correspond-ing generatcorrespond-ing functional with respect toθleads (for small amplitudesη0) to the topological charge correlator
hQ(x)Q(0)i= ΠP(x)/(8π)2
≃ −2¯nδ4(x)−(2¯nγ
η0)
2
hη0(x)η0(0)i.
(18)
The first term is just the pointlike approximation to the direct-instanton contribution evaluated above. The second one is the screening correction, which modifies the nonper-turbative contributions to the pseudoscalar IOPE coefficient into
ΠP(x) = Π( I+ ¯I)
P (x)−(¯nγη0)
2mη0
π2xK1(mη0x). (19)
After implementing finite quark-mass effects (i.e. η0-η8
mixing), the (Minkowski) Borel moments associated with the screening contributions become
R(P,kscr)(τ) =−δk,−1
à F2
η′
m2
η′ + F
2
η
m2
η
!
+Fη2′m
2k η′e
−m2 η′τ+F2
ηm2ηke
−m2
ητ. (20)
The τ-independent term in Eq. (20) is the screening-induced subtraction constant−Π(Pscr)(0). It shows that in-clusion of the screening contributions is mandatory in order to satisfy the axialU(1)Ward identity and the ensuing LET (4). Indeed, direct instantons generate a large subtraction constantΠ(PI+ ¯I)(0) = −27π2n¯ while the LET demands
the zero-momentum limit of the physical correlator to be of the order of the light quark masses, i.e. much smaller. The screening contribution (20) is necessary to cancel most of it and to restore consistency with the LET. In the chiral limit, the screening contribution turns into
Π(Pscr)(0) = F
2
η′
m2
η′
=(16πnγ¯ η0)
2
2¯nγ2
η0
= 27π2¯n (21)
and the cancellation becomes exact (due to the infinite-range interactions mediated by massless Goldstone bosons). The above argument provides compelling evidence for the screening contributions to be an indispensable complement to the direct instantons.
The cancellation between the subtraction terms suggests a simple strategy for renomalizing the screening contribu-tions. Since the large-ρcutoffµ−1amounts to replacingn¯
byn¯dir =ζ¯nwithζ < 1(cf. Eq. (16)), consistency with
the LET requires the same replacement in the screening con-tributions (19).
Besides restoring the axial Ward identity, inclusion of the screening contributions resolves the positivity-bound vi-olation and creates a strong signal for both theη′
and the pseudoscalar glueball resonances in the corresponding Borel sum rules. The screening contributions (20) are of substan-tial size, even relative to the direct-instanton contributions, and they are largest at small and intermediateτ .λ2
scr∼1
GeV2. Moreover, they modify qualitative features of the Borel moments (e.g. the sign of the slope) to which the sum rule fits are very sensitive. Hence, trustworthy sum rule re-sults cannot be obtained even at small and intermediate τ
878 Hilmar Forkel
After including the screening contributions, all four Borel-moment sum rules (9) are stable and yield consistent results. Note that previous analyses of the0−+ sum rules
have discarded thek =−1sum rule and therefore missed the chance to implement the first-principle information from the low-energy theorem, as well as a very useful consistency check. The two-resonance fit is clearly favored over the one-pole approximation, i.e. the IOPE provides clear signals for theη′
resonance (with smallηadmixtures due to mixing) in addition to a considerably heavier0−+ glueball. The0−+
glueball mass is found to bemP = 2.2±0.2GeV, and the
decay constantfP = 0.6±0.25GeV. (For a complete
dis-cussion see [4].)
5
Summary
We have reported recent developments in understanding and evaluating nonperturbative glueball physics in the pseu-doscalar channel on the basis of the operator product ex-pansion [4]. Contrary to naive expectation, much of this nonperturbative physics takes place at surprisingly short dis-tances |x| ∼ 0.2 −0.3 fm, and consequently shows up in the Wilson coefficients of the IOPE. Direct instantons are the paradigm for such physics, and their contributions to the spin-0 glueball correlators are indeed exceptionally large. We have improved on the previous evaluation of these contributions [3] by implementing a realistic instanton size distribution and the renomalization of the instanton-induced coefficients (both of these improvements should be useful in other hadron channels as well). A further new develop-ment is very specific to the0−+glueball channel: we have
found compelling evidence for topological charge screen-ing to provide crucial contributions to the unit operator co-efficient of the pseudoscalar glueball IOPE. The screening contributions form an indispensable complement to the di-rect instantons, roughly speaking “unquenching” them and thereby restoring the axial Ward identity. Moreover, they balance the strong repulsion of the direct instantons, cor-rect the otherwise gross violation of the LET, resolve the violation of the spectral positivity bound and generate a strong (and otherwise absent) signal for the 0−+ glueball
resonance. With screening included, all Borel moment sum rules provide consistent and stable predictions for the fun-damental0+−
glueball properties (mP = 2.2±0.2GeV,
fP = 0.6±0.25GeV).
This work was supported by FAPESP.
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